Question

Find the surface area generated when
y = (x^3/12) + (1/x), from x=1 to x=2 is rotated around the x-axis.

Answer 1

I visualize an infinite number of "rings"

radius of ring = (x^3)/12 + 1/x
circumference = 2π((x^3)/12 + 1/x)

surface area = 2π∫( (x^3)/12 + 1/x) dx from 1 to 2
= 2π [ (x^4)/48 + lnx] from 1 to 2
= 2π (1/3 + ln 2 - 1/48 - ln1)
= 2π(5/16 + ln 2
= π(5/8 + 2ln 2)